What happens if Klefki with "Mischievous Lock" and Empoleon V with "Emperor's Eyes" Ability somehow both become active at the same time, like at the start of the game?
If Abilities activate at the same time, the Abilities of the player whose turn it is activate first.
Source: TPCi Rules Team (2023-04-06)
If Empoleon V with "Emperor's Eyes" Ability is in the Active Spot and my opponent brings up a Klefki with "Mischievous Lock" into their Active Spot, whose abilities are turned off?
Since Emperor's Eyes was in effect before Klefki became active, it takes precedence and Mischievous Lock does not activate as long as Emperor's Eyes is in effect. The converse is also true, if Klefki were active first and Empoleon V is brought up, Emperor's Eyes would not activate.
Source: TPCi Rules Team (2023-04-06)
If my opponent has an Empoleon V with "Emperor's Eyes" Ability active, and I have both an Eevee with a Memory Capsule attached and an Eevee with the "Resonant Evolution" Ability in play, if I evolve the Eevee with a Memory Capsule into a Jolteon with "Thunderous Awakening" Ability can I use "Resonant Evolution" or not?
Yes, that will work because Emperor's Eyes has been disabled.
Source: TPCi Rules Team (2022-10-06)
Does the box regarding taking prizes for KO'ing a Pokémon-ex from the EX-era (like EX Ruby & Sapphire) count as a "Rule Box" for the purpose of Empoleon V's "Emperor's Eyes" Ability?
No, the term Rule Box does not apply to anything prior to Black & White. As of the date of this ruling these things are covered by Rule Box: Pokémon V, Pokémon VMAX, Pokémon-GX, TAG TEAM, Prism Star, Pokémon-EX, Mega Evolution, and Pokémon BREAK (any future cards that may have a Rule Box are also covered.)
Source: TPCi Rules Team (2021-03-18)